Inverse problem of linear combinations of Gaussian convolution kernels (deconvolution) and some applications to proton/photon dosimetry and image processing

The deconvolution of a single Gaussian kernel is extended to a sum of Gaussian kernels with positive coefficients (case 1) and to a Mexican hat (case 2). In case 1 the normalization requires the sum of the normalized Gaussian kernels to be always 1, i.e. c0 + c1 + c2 + = 1. Each coefficient satisfies ck > 0. In case 2 (Mexican hat) the properties c0 + c1 = 1 with c0 > 1and c1 < 0 hold; c1 = 1 ? c0 has to be accounted for the normalization. We discuss examples of the deconvolution of both cases. Case 1 is considered in an analysis of transverse profiles (protons and photons) and in the deconvolution of CT images to eliminate scatter. Case 2 is applied to a proton Bragg curve measured by an ionization chamber and a diode detector. In the domain of the Bragg peak there is a different physical behavior between both measurement methods.

[1]  H. Palmans.  Perturbation factors for cylindrical ionization chambers in proton beams. Part I: corrections for gradients , 2006, Physics in medicine and biology.

[2]  L Paelinck,et al.  The influence of small field sizes, penumbra, spot size and measurement depth on perturbation factors for microionization chambers , 2009, Physics in medicine and biology.

[3]  R. Feynman,et al.  Quantum Mechanics and Path Integrals , 1965 .

[4]  M. Markov.  Invariants and the evolution of nonstationary quantum systems , 1989 .

[5]  C. Case,et al.  MOLIERE'S THEORY OF MULTIPLE SCATTERING. , 1968 .

[6]  Jianhua Yao,et al.  Scale-based scatter correction for computer-aided polyp detection in CT colonography. , 2008, Medical physics.

[7]  G. Molière,et al.  Theorie der Streuung schneller geladener Teilchen , 1955 .

[8]  On the application of Gauss transformation in nonlinear quantum mechanics , 1978 .

[9]  W. Ulmer.  Theoretical aspects of energy–range relations, stopping power and energy straggling of protons , 2007 .

[11]  B. Schaffner.  Proton dose calculation based on in-air fluence measurements , 2008, Physics in medicine and biology.

[12]  J. Damilakis,et al.  Experimental determination of the effect of detector size on profile measurements in narrow photon beams. , 2006, Medical physics.

[13]  Saburou Saitoh.  Approximate real inversion formulas of the gaussian convolution , 2004 .

[14]  W. Ulmer,et al.  A 3D photon superposition/convolution algorithm and its foundation on results of Monte Carlo calculations , 2005, Physics in medicine and biology.

[15]  V. Highland,et al.  Some Practical Remarks on Multiple Scattering , 1975 .

[16]  Wolfgang Gröbner,et al.  Contributions to the method of Lie series , 1967 .

[17]  J. Delgado,et al.  Exact analytical solution of the convolution integral equation for a general profile fitting function and Gaussian detector kernel. , 2000, Physics in medicine and biology.

[18]  J. Delgado,et al.  Experimental determination of the convolution kernel for the study of the spatial response of a detector. , 1998, Medical physics.

[19]  G. Bednarz.  Deconvolution of detector size effect for output factor measurement for narrow Gamma Knife radiosurgery beams , 2000, Proceedings of the 22nd Annual International Conference of the IEEE Engineering in Medicine and Biology Society (Cat. No.00CH37143).

[20]  Analytical and numerical inversion formulas in the Gaussian convolution by using the Paley–Wiener spaces , 2006 .

[21]  E Pedroni,et al.  Experimental characterization and physical modelling of the dose distribution of scanned proton pencil beams , 2005, Physics in medicine and biology.

[22]  I. Das,et al.  Small fields: nonequilibrium radiation dosimetry. , 2007, Medical physics.

[23]  Robert J. Schneider,et al.  Multiple Coulomb scattering of 160 MeV protons , 1993 .

[24]  W. Ulmer.  On the representation of atoms and molecules as self-interacting field with internal structure , 1980 .

[25]  E Pedroni,et al.  Monte Carlo dose calculations for spot scanned proton therapy , 2005, Physics in medicine and biology.

[26]  Deconvolution of detector size effect for output factor measurement for narrow Gamma Knife radiosurgery beams. , 2002 .

[27]  J. Neumann.  Mathematical Foundations of Quantum Mechanics , 1955 .

[28]  W. Ulmer,et al.  The inverse problem of a Gaussian convolution and its application to the finite size of the measurement chambers/detectors in photon and proton dosimetry. , 2003, Physics in medicine and biology.

[29]  Hans A. Bethe,et al.  Moliere's theory of multiple scattering , 1953 .

[30]  Robert D. Nowak,et al.  An EM algorithm for wavelet-based image restoration , 2003, IEEE Trans. Image Process..

[31]  A. Brahme,et al.  Influence of multiple scattering and energy loss straggling on the absorbed dose distributions of therapeutic light ion beams: I. Analytical pencil beam model. , 2004, Physics in medicine and biology.

[32]  S. Saitoh.  APPLICATIONS OF THE REPRODUCING KERNEL THEORY TO INVERSE PROBLEMS , 2001 .